"The ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value . . . appears so simple to us now that we ignore its true merits. But its very simplicity puts our arithmetic in the first rank of useful inventions . . . remember, it escaped the genius of Archimedes and Apollonius." --Laplace
Fundamental mathematical concepts are very difficult to grasp. Although most adults use numbers and mathematics daily in performing activities, the underlying concepts are difficult to learn. Children often require several years to master the premise of recognizing that numbers represent a numerical quantity for a group of real world objects. Further, these abstract numerical quantities can be added and subtracted, which correspond to the number of real objects represented by the numerical quantity.
The grouping of numbers into fields such as hundreds, tens and ones is an abstraction within itself. In the decimal representation, numerical quantities are grouped into sets of singles, tens and hundreds units (and continues into thousands, etc.), allowing any size numerical quantity to be represented with a number. The idea that ten ones is the same as one ten is fairly straightforward, but changes resulting from addition and multiplication can be troublesome. Children must understand that adding single units can affect the tens or even hundreds units of a number.
Subtraction requires an even greater abstraction for children to master. When performing subtraction on two numbers, if the second number has a unit place larger than the first number, the child must "borrow" from the next higher unit of the first number to obtain enough units to perform the subtraction. This concept of borrowing from a higher unit is difficult to grasp. Schools often teach the borrowing process by rote, without allowing the students to truly understand what is going on.
Several teaching blocks and rod systems have been widely used. These sometimes are used for demonstration, but when children use the block systems on their own, there is no feature of the blocks or rods to show a correct solution to a problem. The child can group the blocks or rods in any fashion, with no indication that any particular grouping is better or useful.
Accordingly, what is needed is a system or apparatus allowing children and others to visualize the process of quantifying a set of objects, and once quantified, to manipulate that quantity with various mathematical techniques such as addition and subtraction. The apparatus should also be visually stimulating and exciting to help maintain attention of the users. The apparatus should be fool proof, and allow users to repeat any operation many times and always perform the same steps to get the result, in effect, be self-educating.